If x and y are positive real numbers such that log_x(x^2+12)=4 and 3log_y(x)=1, then x+y equals
Question:
If x and y are positive real numbers such that log_x(x^2+12)=4 and 3log_y(x)=1, then x+y equals
If x and y are positive real numbers such that log_x(x^2+12)=4 and 3log_y(x)=1, then x+y equals
Options
Answer: 10
Explanation:
From log_x(x^2 + 12) = 4 we get x^4 = x^2 + 12. Let t = x^2. Then t^2 - t - 12 = 0 → (t - 4)(t + 3) = 0. Since t > 0, t = 4, so x^2 = 4 → x = 2 (positive). From 3·log_y(x) = 1 we get log_y(x) = 1/3, so y^(1/3) = x → y = x^3. With x = 2, y = 8. Therefore x + y = 2 + 8 = 10. So the correct option is 10.
Explanation:
From log_x(x^2 + 12) = 4 we get x^4 = x^2 + 12. Let t = x^2. Then t^2 - t - 12 = 0 → (t - 4)(t + 3) = 0. Since t > 0, t = 4, so x^2 = 4 → x = 2 (positive). From 3·log_y(x) = 1 we get log_y(x) = 1/3, so y^(1/3) = x → y = x^3. With x = 2, y = 8. Therefore x + y = 2 + 8 = 10. So the correct option is 10.
👉 Want AI explanation? Open in MCQ App
Recommended articles
- How To Cover Current Affairs For Upsc Prelims — Master UPSC Current Affairs with daily newspaper reading, Vision IAS monthly compilations, PT 365, and MCQ practice. Complete strategy guide for Prelims…
- Top 10 Daily Current Affairs Resources For Upsc Aspirants — Discover the best daily current affairs resources for UPSC 2026. Complete guide with The Hindu, PIB, Sansad TV, and 7 more proven sources. Expert…
- Free Cat Previous Year Questions Guide — Master CAT 2025 with free previous year questions. Practice 1000+ PYQs, mock tests, and detailed solutions. Complete guide to CAT preparation using past…